### Abstract

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂_{t}
^{2} − Δ is d’Alembertian. For a given asymptotic profile u_{ap}, we construct a solution u to (NLKG) which converges to u_{ap} as t → ∞. Here the asymptotic profile u_{ap} is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

Original language | English |
---|---|

Pages (from-to) | 8155-8170 |

Number of pages | 16 |

Journal | Transactions of the American Mathematical Society |

Volume | 370 |

Issue number | 11 |

DOIs | |

Publication status | Published - Jan 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

**Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions.** / Masaki, Satoshi; Segata, Junichi.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 370, no. 11, pp. 8155-8170. https://doi.org/10.1090/tran/7262

}

TY - JOUR

T1 - Modified scattering for the quadratic nonlinear Klein–Gordon equation in two dimensions

AU - Masaki, Satoshi

AU - Segata, Junichi

PY - 2018/1/1

Y1 - 2018/1/1

N2 - In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂t 2 − Δ is d’Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

AB - In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein–Gordon equation (NLKG) in two space dimensions: (Formula Presented), where □ = ∂t 2 − Δ is d’Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein–Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on Fourier series expansion of the nonlinearity.

UR - http://www.scopus.com/inward/record.url?scp=85055087517&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85055087517&partnerID=8YFLogxK

U2 - 10.1090/tran/7262

DO - 10.1090/tran/7262

M3 - Article

AN - SCOPUS:85055087517

VL - 370

SP - 8155

EP - 8170

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 11

ER -